Chinese Annals of Mathematics, Series B

, Volume 37, Issue 5, pp 665–682 | Cite as

The cauchy problem for coupled nonlinear Schrödinger equations with linear damping: Local and global existence and blowup of solutions

  • João-Paulo Dias
  • Mário Figueira
  • Vladimir V. Konotop


The authors study, by applying and extending the methods developed by Cazenave (2003), Dias and Figueira (2014), Dias et al. (2014), Glassey (1994–1997), Kato (1987), Ohta and Todorova (2009) and Tsutsumi (1984), the Cauchy problem for a damped coupled system of nonlinear Schrödinger equations and they obtain new results on the local and global existence of H 1-strong solutions and on their possible blowup in the supercritical case and in a special situation, in the critical or supercritical cases.


Nonlinear Schrödinger equations Cauchy problem Blowup of solutions Dissipation 

2000 MR Subject Classification

35Q55 33A05 


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  1. [1]
    Bludov, Yu. V., Driben, R., Konotop, V. V. and Malomed, B. A., Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides, J. Opt., 15, 2013, 064010.CrossRefGoogle Scholar
  2. [2]
    Cazenave, T., Semilinear Schrödinger Equations, Courant Lecture Notes, Vol. 10, Amer. Math. Soc., 2003.CrossRefzbMATHGoogle Scholar
  3. [3]
    Couairon, A. and Mysyrowicz, A., Femtosecond Filamentation in Transparent Media, Phys. Rep., 441, 2007, 47–189.CrossRefGoogle Scholar
  4. [4]
    Dias, J. P. and Figueira, M., On the blowup of solutions of a Schrödinger equation with an inhomogeneous damping coefficient, Comm. Contemp. Math. 16, 2014, 1350036.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Dias, J. P., Figueira, M., Konotop, V. V. and Zezyulin, D. A., Supercritical blowup in coupled paritytime-symmetric nonlinear Schrödinger equations, Studies Appl. Math., 133, 2014, 422–440.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Glassey, R. T., On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18, 1977, 1794–1797.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Jüngel, A. and Weishäupl, R. M., Blow-up in two-component nonlinear Schrödinger systems with an external driven field, Math. Models Meth. Appl. Sciences, 23, 2013, 1699–1727.CrossRefzbMATHGoogle Scholar
  8. [8]
    Kato, T., On nonlinear Schrödinger equations, Ann. Inst. H. PoinCaré Phys. Théor., 46, 1987, 113–129.zbMATHGoogle Scholar
  9. [9]
    Menyuk, C. R., Pulse propagation in an elliptically birefringent medium, IEEE J. Quant. Electron., 25, 1989, 2674.CrossRefGoogle Scholar
  10. [10]
    Ohta, M., and Todorova G., Remarks on global existence and blowup for damped nonlinear Schrödinger equations, Discrete Cont. Dyn. Syst., 23, 2009, 1313–1325.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Pitaevskii, L. and Stringari, S., Bose–Einstein Condensation, Clarendon Press, Oxford, 2003.zbMATHGoogle Scholar
  12. [12]
    Prytula V., Vekslerchik, V. and Pérez-Garcia, V. M., Collapse in coupled nonlinear Schrödinger equations: Sufficient conditions and applications, Physica D, 238, 2009, 1462–1467.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Roberts, D. C. and Newell, A. C., Finite-time collapse of N classical fields described by coupled nonlinear Schrödinger equations, Phys. Rev. E, 74, 2006, 047602.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Tsutsumi, M., Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations, SIAM J. Math. Anal., 15, 1984, 357–366.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • João-Paulo Dias
    • 1
  • Mário Figueira
    • 1
  • Vladimir V. Konotop
    • 2
  1. 1.CMAF-CIO, Faculdade de CiênciasUniversidade de LisboaLisboaPortugal
  2. 2.Centro de Física Teórica e Computacional and Departamento de Física, Faculdade de CiênciasUniversidade de LisboaLisboaPortugal

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